Large language models trained on clinical text risk exposing sensitive patient information, yet differential privacy (DP) methods often severely degrade the diagnostic accuracy needed for deployment. Despite rapid progress in DP optimisation and text generation, it remains unclear which privacy-preserving strategy actually works best for clinical language tasks. We present the first systematic head-to-head comparison of four training pipelines for automated diagnostic coding from hospital discharge summaries. All pipelines use identical 1B-parameter models and matched privacy budgets to predict ICD-9 codes. At moderate and relaxed privacy budgets ($\varepsilon \in \{4, 6\}$), knowledge distillation from DP-trained teachers outperforms both direct DP-SGD and DP-synthetic data training, recovering up to 63\% of the non-private performance whilst maintaining strong empirical privacy (membership-inference AUC $\approx$ 0.5). These findings expose large differences in the privacy-utility trade-off across architectures and identify knowledge distillation as the most practical route to privacy-preserving clinical NLP.